Transcript COMETARY PARALLAX

COMETARY PARALLAX
StarFest 2005
Bays Mountain Preserve
October 22, 2005
John C. Mannone
Abstract
Planetarium software and PowerPoint slide utilities are
engaged to graphically determine the parallax of a near
object observed by amateur astronomers. This graphical
method seems to favorably compare with spherical
trigonometry methods (not discussed). Though applicable
to some planets and our Moon, the technique will be
demonstrated with comets on close approach (~1 au). This
is useful for planned coordinated viewing/photography and
for a classroom experiment to determine distance of
approach. The technique can be extended to very close
objects such as satellites and meteors, but video imaging
and processing will be required.
Definition of Parallax
What is it?
When an object is viewed from two different positions,
there is a shift in the apparent position of the object against
a distant background.
Shift can be caused by several things, e.g.,
Change in refractive index which bends the light
Change in geometry (trigonometric parallax)
(Spectroscopic parallax applies to determination of
distance from spectroscopically determined luminosity and
spectral class)
Trigonometric Parallax
A simple example:
Look at me with one eye shut
Then the other
Note my apparent position against the backdrop is different
Trigonometric Parallax
Eyes are separated some base distance, b
The angular difference of my image perceived by each eye (each
viewing position) is the parallax (angle) related to the base distance
and the my distance to the observer. The further away, the smaller
the angle:
Tycho Brahe tried to apply parallax in 1570’s, but Friedrich Bessel
first successfully applied this to stars in 1838: 61 Cygni: 0.333”
(modern result 0.289”). The closest star, Proxima Centauri, has
largest p = 0.772”
Stellar Parallax
Stellar Parallax
Stellar Parallax
Stellar Parallax
parallax angle
distance
Parallax, p, and distance, d, are related through
simple geometry, especially when the the
parallax is small, as it is in the case of stars.
d (parsec) = 1/p (arcsec)
1 pc = 3.26 ly
Cometary Parallax
Comets approach much closer than stars, so expect parallax angle
be much larger.
Because of its rapid motion (relative to stars), a simultaneous
observation will limit observation to different places on the Earth
(instead of two different orbital positions of the Earth).
This limits the distance between observation sites to the chord
through the Earth connecting the two locations.
A further reduction in the chord because of the comet’s
perspective.
The parallax will be larger only by an order of magnitude
over nearby stars.
Determination of Comet Approach Distance by Parallax
Distance-Parallax Related through the Projected Chord
tan (p/2) = b/2d
d1 = d - R + (R2-b2/4)1/2 ~d for more distant objects
p is the parallax (angle), b is the projected chord distance A”B”
between the 2 observing sites A and B (perpendicular to the zenith line
d1 at a point on the surface of the Earth directly beneath the comet at
C).
Comet’s apparent positions
A”
among background stars
b
R
B”
d1
d
p
C
Determination of Cometary Parallax
Graphical
Software Simulation
Photographic Analysis
Image Overlap/Scaling
Analytical
Three-Dimensional Exact Solution- Celestial Sphere
Spherical Trigonometry
Why the Interest in Cometary Parallax?
I purchased a personally autographed photograph of HaleBopp from Dr. Tom Bopp at UTC in March 2003.
It is one of his favorite photographs by Bill and Sue
Fletcher.
I became interested in everything about the photograph: the
photographic details, identification of the major stars. I
reasoned others might have simultaneously photographed
the comet, especially near closest Earth approach and
wondered if the comet’s distance could be easily determined
by comparing photographs.
Synchronizing time is easy with planetarium software.
Hale-Bopp Trajectory Near Perihelion
Earth Closest Approach: March 22, 1997 (1.315 AU)
Sun Closest Approach: April 1, 1997 03:14 UT (0.914 AU)
“This is the beautiful Comet Hale-Bopp as
it approached Earth in March of 1997.
The solid portion or nucleus of the comet is
made up of ice, frozen gases, dust and small
rock. Compared to most comets Hale-Bopp
is very large - about 35 kilometers in
diameter. As its orbit brought it closer to the
sun, the frozen mass began to melt and a
coma, which is a gaseous cloud, developed
around the nucleus.
This coma has grown to be hundreds of thousands of miles in diameter.
Finally the tail developed which became millions of miles long.
This color photo reveals both the reddish cream-colored dust tail, and the
many long blue streamers of the ion (gas) tail.” (photographers Bill & Sue Fletcher)
TIME Picture of Year 1997, TIME/LIFE Picture of the Century 2000
Joshua Tree National Park
"God just gave me a gift. I get to see things in the sky that the
average person doesn't see…I think that what's out there is
God's creation meant for our enjoyment." Wally Pacholka
Date and Time:
Camera:
Film/Exposure:
Length/Width Ratio:
April 4, 1997, 8 PM PST
50mm Minolta lens f/1.7 on a tripod;
Fuji 800 film (35 mm)/ 30 seconds
1.36 => picture cropped
Joshua Tree located with the help
of digital desert and aviation
charts: Coordinates 34N, 116W
Elevation 3000 to 4000 ft
f = 50 mm, f/ = 1.7, D = 29.4 mm
Approximate FOV:
2arctan [(36 x 24 mm/2)/50 mm]
FOV = 27.0o x 39.6o
Calculated FOV 1.15o x 1.72o
“Comet Hale-Bopp photographed
on the morning of March 8, 1997,
from Stedman, N.C. This 10-minute
exposure was made with a 12.5inch reflecting telescope (f = 1200
mm) and exposed on Fujicolor SG800 Plus film. The telescope
tracked the comet during the
exposure, rendering the stars as
short lines. Hale-Bopp is moving
northward against the stars at the
rate of 1.5 degrees per day*. The
comet continues to be visible to the
naked eye in the predawn
northeastern sky.” (Jim Horne,
photo 33)
~50,000 mph
HALE-BOPP March 8, 1997
9-hour time difference means
photos taken at different local times
Cathedral City, CA, USA
Asagio, Vincenza, Italy
HALE-BOPP March 8, 1997 (actually March 7)
This Fletcher photograph
was made with the special
Schmidt camera/telescope.
An 8-inch Celestron
equivalent to a super fast
(f/1.5) 305 mm telephoto
lens.
Equipped with curved film
holder => no distortion
along width.
Joshua Tree National Park, CA, USA
Wide field of view 4.5o x
6.75o
HALE-BOPP March 7, 1997 4:40 AM
This Fletcher photograph was
made with the special
Schmidt camera/telescope.
Wide field of view 4.5o x 6.75o
An 8-inch Celestron equivalent to a super fast (f/1.5) 305 mm telephoto lens.
Equipped with curved film holder => no distortion along width
Parallax by Graphical Methods
Software Simulation
Photographic Analysis
Parallax is determined by superposition of images with the same
field of view or scale.
Both views are aligned. The transparency can be adjusted with the
picture editing feature. This facilitates the correct overlapping.
Angular separation between the comet and the star is determined (a
standard feature on Starry Night Backyard software).
The parallax is determined by comparison with the scaled cometstar distance.
Hale-Bopp 100 degree field of view from Joshua Tree, California
Hale-Bopp 30 degree field of view from Joshua Tree, California
Hale-Bopp 15 degree field of view from Joshua Tree, California
Hale-Bopp 1 degree field of view from Joshua Tree, California
Hale-Bopp 1 degree field of view from Asagio, Italy
USA
Italian
Italian
USA
Hale-Bopp 1 degree field of view overlays
68% transparency of top slide
USA
Italian
Italian
USA
Hale-Bopp 1 degree field of view overlays
Overlap Background Stars
Hale-Bopp 1 degree field of view overlays
Rotate to align along RA/Dec lines
Italian
Hale-Bopp 1 degree field of view overlays
Re-establish Overlap
Measure length; use ratio and
proportion to obtain parallax
Italian
Measure angular separation
on Starry Night; relate to scale
length
Hale-Bopp 1 degree field of view overlays
Re-establish Overlap
Using a different star, the results
are summarized below
Comet Hale-Bopp
March 8, 1997 11:40Z
Asiago, Italy
0.30 inch parallax
Joshua Tree National Park, CA
Parallax, p = (.30 inch) (249”/10-7/8 inch) = 6.87’’ +/- 10%
Parallax by Analytical Methods
Apparent Comet Positions
Projected on Celestial Sphere
A”
CB
R
b
C
d1
p
B”
Projected Geographic Positions
Celestial Sphere
C
CA
Actual position of comet: C
C
p
CA
DDec
CB
DRA
Apparent positions
of comet from
projected A and B
Parallax seen on a Spherical Triangle
The 3-dimensional Exact Calculation of Parallax
Spherical Geometry
A b
c
C
B
a
Symbols in this
graphic have
different meanings
Parallax is calculated from object’s equatorial coordinates from both
locations using the law of cosines for spherical triangles
cos c = cos a cos b + sin a sin b cos C = sin a' sin b' +cos a' cos b' cos C
c parallax, a and b equatorial colatitudes, C equatorial longitude difference,
a' and b' are the corresponding latitudes = 90-a and 90-b (degrees)
Parallax by Analytical Methods
Three-Dimensional Exact Solution- Celestial Sphere
Spherical Trigonometry
cos p = sin latA sin latB + cos latA cos latB cos (lonB-lonA)
Need chord length to calculate distance
and an understanding of the celestial
rotating coordinate system
Equatorial and Horizon Coordinates
Courtesy of Scott Robert Ladd, “Stellar Cartography”
Greenwich Mean Sidereal Time
Hale-Bopp March 8, 1997 11:40Z
Calculator by AstroJava
“Sidereal time is the measure of the earth's rotation with respect to
distant celestial objects.
By convention, the reference points for Greenwich Sidereal Time
are the Greenwich Meridian and the vernal equinox (the intersection
of the planes of the earth's equator and the earth's orbit, the ecliptic).
The Greenwich sidereal day begins when the vernal equinox is on
the Greenwich Meridian. Greenwich Mean Sidereal Time (GMST)
is the hour angle of the average position of the vernal equinox,
neglecting short term motions of the equinox due to nutation.”
Rick Fisher NRAO Green Bank, WV
Projected Chord Determination
Vector Analysis
or
Coordinate Rotation
Using Transformation Matrices
or
Graphically using a Celestial
Sphere model and string
Not reviewed here
Coordinate Information for Comet Hale-Bopp
March 8, 1997 11:40Z
Simultaneously Viewed from USA and Italy
Hale-Bopp March 8, 1997 11:40Z
Asiago Joshua Tree
Joshua Tree Comet Coordinates
J (now) Epoch from Starry Night Backyard v 3.1
RA 22h 15.348m = a
Dec 39o 49.504’ = d
GST = 22h 44m 51.7s
Lat comet = d = 39.825067o
Lon comet = H = a - GST = -29.514m
@15o/hour H = -7.378417o
Hale-Bopp March 8, 1997 11:40Z
Asiago Joshua Tree
Observer Coordinates (estimated)
A- Joshua Tree
LatA 33o 44.4’ N
LonA 116o 25.2’ W
Time Zone -7 hr => 4:40 am March 8, 1997 local daylight time
B- Asiago
LatB 48o 22.809’ N
LonB 9o 37.331’ E
Time Zone +1 hr => 12:40 am March 9, 1997 local standard time
Lat comet = d = 39.825067o
Lon comet = H = -7.378417o
(from Joshua)
Hale-Bopp March 8, 1997 11:40Z
Asiago Joshua Tree
Actual distance to Earth 1.382 AU
From orbital parameters in Starry Night
b = 6672.88 km from spherical trigonometry (compare to
Earth radius of 6378 km)
p = 6.87” from graphical method
d1 =1.372 AU (0.72% high)
Accurate, but imprecise (10%)
p = 6.8319” from spherical trigonometry
d1 = 1.385 AU (0.19% high)
Accurate and precise
d1 = (b/2)cotan(p/2)
Conclusion
1) Graphical determination of parallax is effective with
planetarium software, such as Starry Night, and PowerPoint
picture options. Scanned photographs of simultaneous
photographs would be analyzed in the same way.
2) Results are very accurate, though more difficult to reproduce
than with spherical trigonometry. This was applied to Comet
Hyakutake with superior results.
3) Procedure is sufficiently simple for secondary educational
outreach and amateur astronomy, yet easily extended to
collegiate level.
4) Extension to Lunar parallax using solar system objects like
Jupiter as background is very effective.
Conclusion
5) Extension to ISS is possible with the help of Heaven-Above
website for satellite position and altitude. Video imaging and
processing would be required to synchronize simultaneous
observations. This would be a good calibration technique since
the distance to the satellite would be known.
6) Extension to Meteoritic parallax is an advanced experiment
similar to satellite tracking except for the uncertainty of when a
rapidly moving meteor will appear. It’s height is unknown, but is
in the ionosphere and could be determined.